Abstract: The reproducing kernel Hilbert space (RKHS) method is arguably the most popular approach in machine learning to deal with nonlinearity in data. However, it has not been as widely adopted in statistical analyses as might be expected. One reason is that its statistical properties have not yet been adequately understood and, in particular, its asymptotic inference. In this paper, we introduce a symmetric periodic Gaussian kernel and show, in the generic regression setting where the regression function is in the Sobolev spaces, that the method under consideration is asymptotically normal. This asymptotic distribution also provides an explanation to the estimation efficiency of regularization method using the Gaussian reproducing kernel. We include simulation results to illustrate the finite sample properties of the method.

Key words and phrases: Asymptotic normality, estimation efficiency, Gaussian reproducing kernel, nonparametric estimation, sobolev space.